Chapter 4,5

Time Value of Money

Learning Goals1. Understand the concept of future value, their

calculation for a single amount, and the

relationship of present to future cash flow.

2. Find the future value and present value of both

an ordinary annuity and an annuity due, and

the present value of a perpetuity.

Learning Goals

4. Calculate the present value of a mixed stream of cash

flows.*

5. Understand the effect that compounding more

frequently than annually has on future value and the

effective annual interest rate.

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Question?

Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 after one year, or one that would return

$220,000 after two years?

Answer!

It depends on the interest rate!

The Role of Time Value in Finance

• Most financial decisions involve costs & benefits that

are spread out over time.

• Time value of money allows comparison of cash flows

from different periods.

Basic Concepts

• Future Value: compounding or growth over time

• Present Value: discounting to today’s value

• Single cash flows & series of cash flows can be

considered

• Time lines are used to illustrate these relationships

Computational Aids

• Use the Equations

• Use the Financial Tables

• Use Financial Calculators

• Use Spreadsheets

Computational AidsTime Line

Computational AidsFinancial Calculators

Basic Patterns of Cash Flow

• The cash inflows and outflows of a firm can be

described by its general pattern.

• The three basic patterns include a single amount, an

annuity, or a mixed stream:

Simple Interest

• Year 1: 5% of $100 = $5 + $100 = $105

• Year 2: 5% of $100 = $5 + $105 = $110

• Year 3: 5% of $100 = $5 + $110 = $115

• Year 4: 5% of $100 = $5 + $115 = $120

• Year 5: 5% of $100 = $5 + $120 = $125

With simple interest, you don’t earn interest on interest.

Compound Interest

• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00

• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25

• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76

• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55

• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

With compound interest, a depositor earns interest on interest!

Time Value Terms

• PV0 = present value or beginning amount

• k = interest rate

• FVn = future value at end of “n” periods

• n = number of compounding periods

• A = an annuity (series of equal payments or

receipts)

Four Basic Models

• FVn = PV0(1+k)n = PV(FVIFk,n)

• PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)

Future Value Example

You deposit $2,000 today at 6%

interest. How much will you have in 5

years?

$2,000 x (1.06)5 = $2,000 x 1.3382 = $2,676.40

Algebraically and Using FVIF Tables

Nominal & Effective Rates• The nominal interest rate is the stated or contractual rate of

interest charged by a lender or promised by a borrower.

• The effective interest rate is the rate actually paid or earned.

• In general, the effective rate > nominal rate whenever

compounding occurs more than once per year

EAR = (1 + k/m) m -1

Nominal & Effective Rates

• For example, what is the effective rate of interest on

your credit card if the nominal rate is 18% per year,

compounded monthly?

EAR = (1 + .18/12) 12 -1

EAR = 19.56%

Present Value• Present value is the current dollar value of a future

amount of money.

• It is based on the idea that a dollar today is worth

more than a dollar tomorrow.

• It is the amount today that must be invested at a given

rate to reach a future amount.

• Calculating present value is also known as

discounting.

• The discount rate is often also referred to as the

opportunity cost, the discount rate, the required return,

and the cost of capital.

Present Value Example

How much must you deposit today in order to

have $2,000 in 5 years if you can earn 6%

interest on your deposit?

$2,000 x [1/(1.06)5] = $2,000 x 0.74758 =

$1,494.52

Algebraically and Using PVIF Tables

Present Value of a Perpetuity

• A perpetuity is a special kind of annuity.

• With a perpetuity, the periodic annuity or cash flow

stream continues forever.

PV = Annuity/k

• For example, how much would I have to deposit today in

order to withdraw $1,000 each year forever if I can earn

8% on my deposit?

PV = $1,000/.08 = $12,500

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for

three years.

FVA = 100(FVIFA,5%,3) = $315.25

Year 1 $100 deposited at end of year = $100.00

Year 2 $100 x .05 = $5.00 + $100 + $100 = $205.00

Year 3 $205 x .05 = $10.25 + $205 + $100 = $315.25

Using the FVIFA Tables

Future Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much will your deposits grow to if you

deposit $100 at the end of each year at 5% interest for

three years.

Using Calculator/Excel

PMT 100$ k 5.0%n 3FV? 315.25$

Excel Function

=FV (interest, periods, pmt, PV)

=FV (.06, 5,100, )

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could

afford annual payments of $2,000 (which includes

both principal and interest) at the end of each year for

three years at 10% interest?

PVA = 2,000(PVIFA,10%,3) = $4,973.70

Using PVIFA Tables

Present Value of an Ordinary Annuity

• Annuity = Equal Annual Series of Cash Flows

• Example: How much could you borrow if you could

afford annual payments of $2,000 (which includes

both principal and interest) at the end of each year for

three years at 10% interest?

Using Excel

PMT 2,000$ I 10.0%n 3PV? $4,973.70

Excel Function

=PV (interest, periods, pmt, FV)

=PV (.10, 3, 2000, )

Present Value of a Mixed Stream

• A mixed stream of cash flows reflects no particular

pattern

• Find the present value of the following mixed stream

assuming a required return of 9%.

Using Tables

Year Cash Flow PVIF9%,N PV

1 400 0.917 366.80$

2 800 0.842 673.60$

3 500 0.772 386.00$

4 400 0.708 283.20$

5 300 0.650 195.00$

PV 1,904.60$